A straight line passes through the point of Intersection of the lines `x- 2y - 2 = 0 and 2x - by - 6= 0` and the origin, then the set of values of 'b' for which the acute angle between this line and `y=0` is less than `45^@` is

Options

A. A. `(-oo,4)uu(7,oo)`
B. B. `(-oo,5)uu(7,oo)`
C. C. `(-oo,4)uu(5,7)uu(7,oo)`
D. D. `(-oo,4)uu(4,5)uu(7,oo)`

Answer

D

Explanation

Line passes through the point of intersection of `x - 2y - 2 = 0` and `2x - by - 6 = 0`
So, its equation is given by `lambda (x-2y -2) +(2x -by -6) =0` As it passes through the origin
`- 2 lambda -6 = 0 rArrlambda =- 3`
`:.` Equation of the line is `-x +(6-b)y =0`
Its slope is `(1)/(6-b)`
As its angle with `y =0` is less than `(pi)/(4)`
`:. -1 lt (1)/(6-b) lt 1`
`rArr 6 - b gt 1` or `lt -1`
`rArr b lt 5` or `b gt 7`
But `b ne 4` (as the lines intersect)
`:. b in (-oo,4) uu (4,5) uu (7,oo)`